We have been to Yellowstone and Grand Teton five times since 2005. Planning to return again this year. Our favorite time to visit is the first two weeks of June. We travel without advance reservations, so beating the crowds that arrive after kids get out of school works for us. For Yellowstone we have been successful getting a site by calling a day or two ahead for the campgrounds that take reservations or showing up in the AM at the first come-first served campgrounds. We don't need hookups every night, which makes things a lot easier.
For Grand Teton, we use Gros Ventre campground 90% of the time. It hardly ever fills - never in early June.
We don't have a toad and drive our 24 ft C around the Parks. After the second week of June parking does get to be a problem in Yellowstone.
We have been snowed on a couple of times in mid-June. Melts quick though.
Our cell (Verizon) works at Mammoth Hot Springs and Old Faithful. I think there is cell service at Grant Village. No problem with cell coverage in Grand Teton in the valley.